Quadratic Differentials Research Articles

The Mother Body Phase Transition in the Normal Matrix Model

The normal matrix model with algebraic potential has gained a lot of attention recently, partially in virtue of its connection to several other topics as quadrature domains, inverse potential problems and the Laplacian growth. In this paper we consider the normal matrix model with cubic plus linear potential. To regularize the model, we follow Elbau & Felder and introduce a cut-off. In the large size limit, the eigenvalues of the model accumulate uniformly within a certain domain $\Omega$ that we determine explicitly by finding the rational parametrization of its boundary. We also study in details the mother body problem associated to $\Omega$. It turns out that the mother body measure $\mu_*$ displays a novel phase transition that we call the mother body phase transition: although $\partial \Omega$ evolves analytically, the mother body measure undergoes a one-cut to three-cut phase transition. To construct the mother body measure, we define a quadratic differential $\varpi$ on the associated spectral curve, and embed $\mu_*$ into its critical graph. Using deformation techniques for quadratic differentials, we are able to get precise information on $\mu_*$. In particular, this allows us to determine the phase diagram for the mother body phase transition explicitly. Following previous works of Bleher & Kuijlaars and Kuijlaars & Lopez, we consider multiple orthogonal polynomials associated with the model. Applying the Deift-Zhou nonlinear steepest descent method to the associated Riemann-Hilbert problem, we obtain asymptotic formulas for these polynomials. Due to the presence of the linear term in the potential, there are no rotational symmetries in the model. This makes the construction of the associated $g$-functions significantly more involved, and the critical graph of $\varpi$ becomes the key technical tool in this analysis as well.

Supercritical regime for the kissing polynomials

We study a family of polynomials which are orthogonal with respect to the varying, highly oscillatory complex weight function eniλz on [−1,1], where λ is a positive parameter. This family of polynomials has appeared in the literature recently in connection with complex quadrature rules, and their asymptotics have been previously studied when λ is smaller than a certain critical value, λc. Our main goal is to compute their asymptotics when λ>λc.We first provide a geometric description, based on the theory of quadratic differentials, of the curves in the complex plane which will eventually support the asymptotic zero distribution of these polynomials. Next, using the powerful Riemann–Hilbert formulation of the orthogonal polynomials due to Fokas, Its, and Kitaev, along with its method of asymptotic solution via Deift–Zhou nonlinear steepest descent, we provide uniform asymptotics of the polynomials throughout the complex plane.Although much of this asymptotic analysis follows along the lines of previous works in the literature, the main obstacle appears in the construction of the so-called global parametrix. This construction is carried out in an explicit way with the help of certain integrals of elliptic type. In stark contrast to the situation one typically encounters in the presence of real orthogonality, an interesting byproduct of this construction is that there is a discrete set of values of λ for which one cannot solve the model Riemann–Hilbert problem, and as such the corresponding polynomials fail to exist.

Surfaces in pseudo-Riemannian space forms with zero mean curvature vector

We characterize a space-like surface in a pseudo-Riemannian space form with zero mean curvature vector, in terms of complex quadratic differentials on the surface as sections of a holomorphic line bundle. In addition, combining them, we have a holomorphic quartic differential. If the ambient space is $S^4$, then this differential is just one given in [5]. If the space is $S^4_1$, then the differential coincides with a holomorphic quartic differential in [6] on a Willmore surface in $S^3$ corresponding to the original surface through the conformal Gauss map. We define the conformal Gauss maps of surfaces in $E^3$ and $H^3$, and space-like surfaces in $S^3_1$, $E^3_1$, $H^3_1$ and the cone of future-directed light-like vectors of $E^4_1$, and have results which are analogous to those for the conformal Gauss map of a surface in $S^3$.

Solutions to Donaldson’s Hyperkähler Reduction on a Curve

We study an infinite-dimensional hyperkahler reduction introduced by Donaldson and associated with the constant scalar curvature equation on a Riemann surface. It is known that the corresponding moment map equations admit special solutions constructed from holomorphic quadratic differentials. Here we obtain a more general existence result and so a larger hyperkahler moduli space.

Global Topological Configurations of Singularities for the Whole Family of Quadratic Differential Systems

In Artes et al. (Geometric configurations of singularities of planar polynomial differential systems. A global classification in the quadratic case. Birkhauser, Basel, 2019) the authors proved that there are 1765 different global geometrical configurations of singularities of quadratic differential systems in the plane. There are other 8 configurations conjectured impossible, all of them related with a single configuration of finite singularities. This classification is completely algebraic and done in terms of invariant polynomials and it is finer than the classification of quadratic systems according to the topological classification of the global configurations of singularities, the goal of this article. The long term project is the classification of phase portraits of all quadratic systems under topological equivalence. A first step in this direction is to obtain the classification of quadratic systems under topological equivalence of local phase portraits around singularities. In this paper we extract the local topological information around all singularities from the 1765 geometric equivalence classes. We prove that there are exactly 208 topologically distinct global topological configurations of singularities for the whole quadratic class. The 8 global geometrical configurations conjectured impossible do not affect this number of 208. From here the next goal would be to obtain a bound for the number of possible different phase portraits, modulo limit cycles.

Masur–Veech volumes, frequencies of simple closed geodesics, and intersection numbers of moduli spaces of curves

We express the Masur–Veech volume and the area Siegel–Veech constant of the moduli space Qg,n of genus g meromorphic quadratic differentials with at most n simple poles and no other poles as polynomials in the intersection numbers ∫M‾ g′,n′ψ1d1⋯ψ n′dn′ with explicit rational coefficients, where g′<g and n′<2g+n. The formulas obtained in this article are derived from lattice point counts involving the Kontsevich volume polynomials N g′,n′(b1,…,bn′) that also appear in Mirzakhani’s recursion for the Weil–Petersson volumes of the moduli spaces Mg′,n′(b1,…,bn′) of bordered hyperbolic surfaces with geodesic boundaries of lengths b1,…,bn′. A similar formula for the Masur–Veech volume (but without explicit evaluation) was obtained earlier by Mirzakhani through a completely different approach. We prove a further result: the density of the mapping class group orbit Modg,n⋅γ of any simple closed multicurve γ inside the ambient set MLg,n(Z) of integral measured laminations, computed by Mirzakhani, coincides with the density of square-tiled surfaces having horizontal cylinder decomposition associated to γ among all square-tiled surfaces in Qg,n. We study the resulting densities (or, equivalently, volume contributions) in more detail in the special case when n=0. In particular, we compute explicitly the asymptotic frequencies of separating and nonseparating simple closed geodesics on a closed hyperbolic surface of genus g for all small genera g, and we show that in large genera the separating closed geodesics are 2 3πg⋅1 4g times less frequent.

Boundary Behaviour of some Conformal Invariants on Planar Domains

The purpose of this note is to use the scaling principle to study the boundary behaviour of some conformal invariants on planar domains. The focus is on the Aumannâ��CarathA©odory rigidity constant, the higher order curvatures of the CarathA©odory metric and a conformal metric arising from holomorphic quadratic differentials.

Regular globally hyperbolic maximal anti‐de Sitter structures

Let Σ be a connected, oriented surface with punctures and negative Euler characteristic. We introduce regular globally hyperbolic anti-de Sitter structures on Σ × R and provide two parameterisations of their deformation space: as an enhanced product of two copies of the Fricke space of Σ and as the bundle over the Teichmüller space of Σ whose fibre consists of meromorphic quadratic differentials with poles of order at most 2 at the punctures.

ENUMERATION OF MEANDERS AND MASUR–VEECH VOLUMES

A meander is a topological configuration of a line and a simple closed curve in the plane (or a pair of simple closed curves on the 2-sphere) intersecting transversally. Meanders can be traced back to H. Poincaré and naturally appear in various areas of mathematics, theoretical physics and computational biology (in particular, they provide a model of polymer folding). Enumeration of meanders is an important open problem. The number of meanders with $2N$ crossings grows exponentially when $N$ grows, but the long-standing problem on the precise asymptotics is still out of reach. We show that the situation becomes more tractable if one additionally fixes the topological type (or the total number of minimal arcs) of a meander. Then we are able to derive simple asymptotic formulas for the numbers of meanders as $N$ tends to infinity. We also compute the asymptotic probability of getting a simple closed curve on a sphere by identifying the endpoints of two arc systems (one on each of the two hemispheres) along the common equator. The new tools we bring to bear are based on interpretation of meanders as square-tiled surfaces with one horizontal and one vertical cylinder. The proofs combine recent results on Masur–Veech volumes of moduli spaces of meromorphic quadratic differentials in genus zero with our new observation that horizontal and vertical separatrix diagrams of integer quadratic differentials are asymptotically uncorrelated. The additional combinatorial constraints we impose in this article yield explicit polynomial asymptotics.

Diffusion Rate of Windtree Models and Lyapunov Exponents

Consider a windtree model with one or several right-angled obstacles placed periodically on the plan. We show that their diffusion rate is associated to the largest Lyapunov exponent of some quadratic differentials stratum, and exhibit a general strategy to associate to a windtree model a Lyapunov exponent corresponding to its diffusion rate. This results enables us to compute numerically these diffusion rates, and to observe via computer experiments its behaviour according to the shape of the obstacles.

Contribution of one-cylinder square-tiled surfaces to Masur-Veech volumes

We compute explicitly the absolute contribution of square-tiled surfaces having a single horizontal cylinder to the Masur-Veech volume of any ambient stratum of Abelian differentials. The resulting count is particularly simple and efficient in the large genus asymptotics. Using the recent results of Aggarwal and of Chen-Moeller-Zagier on the long-standing conjecture about the large genus asymptotics of Masur-Veech volumes, we derive that the relative contribution is asymptotically of the order 1/d, where d is the dimension of the stratum. Similarly, we evaluate the contribution of one-cylinder square-tiled surfaces to Masur-Veech volumes of low-dimensional strata in the moduli space of quadratic differentials. We combine this count with our recent result on equidistribution of one-cylinder square-tiled surfaces translated to the language of interval exchange transformations to compute empirically approximate values of the Masur-Veech volumes of strata of quadratic differentials of all small dimensions.

Contribution of one-cylinder square-tiled surfaces to Masur-Veech volumes

We compute explicitly the absolute contribution of square-tiled surfaces having a single horizontal cylinder to the Masur-Veech volume of any ambient stratum of Abelian differentials. The resulting count is particularly simple and efficient in the large genus asymptotics. Using the recent results of Aggarwal and of Chen-Moeller-Zagier on the long-standing conjecture about the large genus asymptotics of Masur-Veech volumes, we derive that the relative contribution is asymptotically of the order 1/d, where d is the dimension of the stratum. Similarly, we evaluate the contribution of one-cylinder square-tiled surfaces to Masur-Veech volumes of low-dimensional strata in the moduli space of quadratic differentials. We combine this count with our recent result on equidistribution of one-cylinder square-tiled surfaces translated to the language of interval exchange transformations to compute empirically approximate values of the Masur-Veech volumes of strata of quadratic differentials of all small dimensions.

Pluripotential theory on Teichmüller space I: Pluricomplex Green function

This is the first paper in a series of investigations of the pluripotential theory on Teichmüller space. One of the main purposes of this paper is to give an alternative approach to the Krushkal formula of the pluricomplex Green function on Teichmüller space. We also show that Teichmüller space carries a natural stratified structure of real-analytic submanifolds defined from the structure of singularities of the initial differentials of the Teichmüller mappings from a given point. We will also give a description of the Levi form of the pluricomplex Green function using the Thurston symplectic form via Dumas’ symplectic structure on the space of holomorphic quadratic differentials.

Cluster exchange groupoids and framed quadratic differentials

We introduce the cluster exchange groupoid associated to a non-degenerate quiver with potential, as an enhancement of the cluster exchange graph. In the case that arises from an (unpunctured) marked surface, where the exchange graph is modelled on the graph of triangulations of the marked surface, we show that the universal cover of this groupoid can be constructed using the covering graph of triangulations of the surface with extra decorations. This covering graph is a skeleton for a space of suitably framed quadratic differentials on the surface, which in turn models the space of Bridgeland stability conditions for the 3-Calabi-Yau category associated to the marked surface. By showing that the relations in the covering groupoid are homotopically trivial when interpreted as loops in the space of stability conditions, we show that this space is simply connected.

Holomorphic quadratic differentials on graphs and the chromatic polynomial

We study “holomorphic quadratic differentials” on graphs. We relate them to the reactive power in an LC circuit, and also to the chromatic polynomial of a graph. Specifically, we show that the chromatic polynomial χ of a graph G, at negative integer values, can be evaluated as the degree of a certain rational mapping, arising from the defining equations for a holomorphic quadratic differential. This allows us to give an explicit integral expression for χ(−k).

Recurrence of quadratic differentials for harmonic measure

AbstractWe consider random walks on the mapping class group that have finite first moment with respect to the word metric, whose support generates a non-elementary subgroup and contains a pseudo-Anosov map whose invariant Teichmüller geodesic is in the principal stratum of quadratic differentials. We show that a Teichmüller geodesic typical with respect to the harmonic measure for such random walks, is recurrent to the thick part of the principal stratum. As a consequence, the vertical foliation of such a random Teichmüller geodesic has no saddle connections.

Polynomial quadratic differentials on the complex plane and light-like polygons in the Einstein Universe

We construct geometrically a homeomorphism between the moduli space of polynomial quadratic differentials on the complex plane and light-like polygons in the 2-dimensional Einstein Universe. As an application, we find a class of minimal Lagrangian maps between ideal polygons in the hyperbolic plane.

Harmonic maps and wild Teichmüller spaces

We use meromorphic quadratic differentials with higher order poles to parametrize the Teichmüller space of crowned hyperbolic surfaces. Such a surface is obtained on uniformizing a compact Riemann surface with marked points on its boundary components, and has noncompact ends with boundary cusps. This extends Wolf’s parametrization of the Teichmüller space of a closed surface using holomorphic quadratic differentials. Our proof involves showing the existence of a harmonic map from a punctured Riemann surface to a crowned hyperbolic surface, with prescribed principal parts of its Hopf differential which determine the geometry of the map near the punctures.

Principal boundary of moduli spaces of abelian and quadratic differentials

The seminal work of Eskin–Masur–Zorich described the principal boundary of moduli spaces of abelian differentials that parameterizes flat surfaces with a prescribed generic configuration of short parallel saddle connections. In this paper we describe the principal boundary for each configuration in terms of twisted differentials over Deligne–Mumford pointed stable curves. We also describe similarly the principal boundary of moduli spaces of quadratic differentials originally studied by Masur–Zorich. Our main technique is the flat geometric degeneration and smoothing developed by Bainbridge–Chen–Gendron–Grushevsky–Möller.

Отпечатки, лемнискаты и квадратичные дифференциалы

We discuss some aspects of the theory of recognition of two-dimensional shapes by means of fingerprints of Jordan curves. An interesting approach to problems on shape recognition suggested by P.~Ebenfelt, D.~Khavinson, and H.~Shapiro and extended further by M.~Younsi reveals the fact that the fingerprints of polynomial lemniscates and, more generally, fingerprints of rational lemniscates can be obtained as solutions to certain functional equations involving Blaschke products. Our main goal here is to develop an approach which relates fingerprints of Jordan curves composed of arcs of trajectories and orthogonal trajectories of certain quadratic differentials with solutions of functional equations involving pullbacks of these quadratic differentials under appropriate Riemann mapping functions. In particular, we show that the previous results of P.~Ebenfelt, D.~Khavinson, and H.~Shapiro and the recent results of M.~Younsi follow from our more general theorems as special cases.